# PROBABILISTIC COMBINATORIAL OPTIMIZATION: MOMENTS,

### Transcript of PROBABILISTIC COMBINATORIAL OPTIMIZATION: MOMENTS,

PROBABILISTIC COMBINATORIAL OPTIMIZATION: MOMENTS,SEMIDEFINITE PROGRAMMING, AND ASYMPTOTIC BOUNDS∗

DIMITRIS BERTSIMAS† , KARTHIK NATARAJAN‡ , AND CHUNG-PIAW TEO§

SIAM J. OPTIM. c© 2004 Society for Industrial and Applied MathematicsVol. 15, No. 1, pp. 185–209

Abstract. We address the problem of evaluating the expected optimal objective value of a0-1 optimization problem under uncertainty in the objective coefficients. The probabilistic model weconsider prescribes limited marginal distribution information for the objective coefficients in the formof moments. We show that for a fairly general class of marginal information, a tight upper (lower)bound on the expected optimal objective value of a 0-1 maximization (minimization) problem can becomputed in polynomial time if the corresponding deterministic problem is solvable in polynomialtime. We provide an efficiently solvable semidefinite programming formulation to compute this tightbound. We also analyze the asymptotic behavior of a general class of combinatorial problems thatincludes the linear assignment, spanning tree, and traveling salesman problems, under knowledge ofcomplete marginal distributions, with and without independence. We calculate the limiting constantsexactly.

Key words. combinatorial optimization, probabilistic analysis, convex optimization, momentsproblem

AMS subject classifications. 90C22, 90C27

DOI. 10.1137/S1052623403430610

1. Introduction. We analyze the optimal objective value of the generic 0-1 op-timization problem with random objective coefficients. Let X = 1, . . . , n and Pbe a nonnegative integer that denotes the number of feasible solutions to the com-binatorial optimization problem. Let Bp, p = 1, . . . , P be nonempty subsets of Xthat correspond to the set of feasible solutions. Without loss of generality, we assumethat the problem does not contain redundant variables, namely, for each i ∈ X, thereexists at least one feasible solution that contains i and at least one feasible solutionthat does not contain i. The nominal 0-1 optimization problem in maximization formis

Z∗max(c) = max

p=1,...,P

∑i∈Bp

ci.(1.1)

Traditional models solve the nominal 0-1 optimization problem under the assump-tion that the objective vector c is completely deterministic. However, such modelsignore the inherent data uncertainty that may affect the quality of the output so-lution. To overcome this shortcoming, an increasing research effort has focused onaddressing uncertainty in the objective function. The standard approach to tacklesuch uncertainty is to assume an underlying distribution for the objective vector c.Let θ represent this prescribed multivariate distribution. For example, when θ is spec-ified to be uniform on [0, 1] and i.i.d. for each coefficient, it reduces to the classical

∗Received by the editors June 24, 2003; accepted for publication (in revised form) March 25,2004; published electronically November 17, 2004. This research was partially supported by theSingapore-MIT Alliance.

http://www.siam.org/journals/siopt/15-1/43061.html†Boeing Professor of Operations Research, Sloan School of Management and Operations Research

Center, Massachusetts Institute of Technology, E53-363, Cambridge, MA 02139 ([email protected]).‡High Performance Computation for Engineered Systems, Singapore-MIT Alliance, Singapore

119260 (karthik [email protected]).§Department of Decision Sciences, NUS Business School, National University of Singapore, Sin-

gapore 117591 ([email protected]).

185

186 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

model studied in probabilistic combinatorial optimization [28]. Examples of problemsthat have been studied under this model are the linear assignment [1], quadratic as-signment [6], and spanning tree problems [10]. The primary focus of such techniquesis to estimate the expected optimal objective value:

Eθ[Z∗max(c)].(1.2)

Evaluating this expectation exactly is nontrivial. Moreover, computing bounds onthis expected optimal value is already quite challenging.

In this paper, we study this problem under a more general setting. Specifically,we consider a class of feasible distributions Θ, and our objective is to find a tightupper bound on the expected optimal objective value of the nominal 0-1 maximizationproblem:

Z∗max = supθ Eθ[Z

∗max(c)]

s.t. θ ∈ Θ.(1.3)

By a tight upper bound we refer to a valid upper bound on the expected objectivevalue that is achieved either exactly or asymptotically by a set of feasible distribu-tions. Problem (1.3) arises naturally in applications where the distribution for theobjective coefficients is not completely known. As an example, consider the problemof estimating an upper bound on the expected time of completion of a project thatconsists of activities with precedence relationships for which we have information onthe expected duration and variability of individual activities but no information ontheir correlation (see also section 4). Specifying the exact distribution in such prob-lems is a potentially difficult task. Most traditional approaches make the additionalassumption of independence and try to estimate the expected optimal value. We,however, drop this assumption of independence completely and provide insights intothe performance of the problem under dependence. Obtaining good lower boundson Eθ[Z

∗max(c)] is relatively easier by using Jensen’s inequality. Our results can be

easily extended to the analogous problem of finding a tight lower bound Z∗min on the

expected optimal objective value of the 0-1 minimization problem by setting c := −c.

1.1. Structure of data uncertainty. The feasible set of multivariate distri-butions Θ we consider is characterized by known information on the marginal dis-tributions of each objective coefficient. Let Fi(c) = P(ci ≤ c) denote the completemarginal distribution function of ci. We let Θ(F1, . . . , Fn) denote the set of multivari-ate distributions compatible with the marginal distributions Fi. Note that this modeldoes not assume that objective coefficients are independently distributed, and henceΘ is not uniquely specified. Under complete knowledge of marginal distributions Fi,problem (1.3) is formulated as

Z∗max = supθ Eθ[Z

∗max(c)]

s.t. θ ∈ Θ(F1, . . . , Fn).(1.4)

Motivated by an application in project management, problem (1.4) was studied byMeilijson and Nadas [21] in the context of computing the longest path in a directedacyclic graph. Their main result shows that Z∗

max can be computed by solving aconvex minimization reformulation of this problem. Weiss [31] observed that thisapproach generalizes to the maximum flow and shortest route problem.

In this paper, we generalize this result to problems with limited information onthe distribution Fi. We assume that the objective coefficient ci takes value in Ωi (i.e.,

PROBABILISTIC COMBINATORIAL OPTIMIZATION 187

the support of the random variable is possibly a subset of Ωi). The distribution isalso assumed to satisfy the moment equality constraints on real-valued functions ofci in the form EFi

[fik(ci)] = mik, k = 1, . . . , ki. In the classical case, with powerfunctions fik(ci) = cki , this reduces to knowing the first ki moments of ci. Given thisinformation, the feasible set of marginal distributions for coefficient ci is representedas

Fi =Fi

∣∣∣ EFi[fik(ci)] = mik, k = 1, . . . , ki, EFi

[IΩi] = 1

,(1.5)

where IS(ci) = 1 if ci ∈ S and 0 otherwise. We assume that the feasible region Fi isnonempty and EFi

[|ci|

]< ∞ to guarantee a finite value for Z∗

max. Under the marginalmoment information model, problem (1.3) is formulated as

Z∗max = supθ,F1,...,Fn

Eθ[Z∗max(c)]

s.t. Fi ∈ Fi, i = 1, . . . , n,

θ ∈ Θ(F1, . . . , Fn).

(1.6)

Remark. We make no assumptions on cross moment information in Θ. As thesize of the problem increases, the number of estimates needed to capture dependentstructures grows exponentially in the dimension of the problem. Instead of trying tocharacterize this exact structure, we optimize over all dependent structures with thegiven marginal moment information.

1.2. Contributions. In this paper, building on the work of Bertsimas andPopescu [4] connecting moment problems and semidefinite optimization, we gener-alize the approach by Meilijson and Nadas [21] and develop techniques to computeZ∗

max and Z∗min for general 0-1 optimization problems. Our main contributions are as

follows:(a) We provide a general optimization formulation to compute Z∗

max under limitedinformation on the marginal distributions of the objective coefficients. Thisformulation has an exponential number of constraints in general.

(b) Given limited marginal moment information for the objective coefficients, weshow that an upper bound on Z∗

max, first proposed in [21] in the contextof the longest path problem in a directed acyclic graph, is tight for generalcombinatorial problems. Note that the tightness of the bound even in thecontext of the longest path problem has not been known to hold in general;see the discussion in [5, p. 844].

(c) For piecewise polynomial functions fik, we show that Z∗max can be computed

in polynomial time if the nominal 0-1 maximization problem can be solved inpolynomial time. We provide an efficiently solvable semidefinite programmingformulation with which to compute the Z∗

max in this case.(d) We characterize the asymptotic behavior of the proposed bounds for a general

class of combinatorial problems under knowledge of complete and identicalmarginal distributions. No assumptions on the independence of distributionsis made. For this model, we show that for the minimum cost N × N linearassignment problem, the spanning tree, and the traveling salesman problemon N vertices, with the distribution function of the cost coefficients satisfyingFα(c) = ραc

α as c ↓ 0 for α ≥ 0, the tight lower bound Z∗min scales as

limN→∞

(Z∗

min

N (α−1)/α

)= C∗

α,

188 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

Table 1.1

Scaling constants with distribution function of form F (c) = ραcα as c ↓ 0.

0-1 Optimization problem limN→∞

(Z∗

min

N(α−1)/α

)

N ×N Linear assignment(

αα+1

)ρ−1/αα

Spanning tree on N node complete graph(

αα+1

)ρ−1/αα 21/α

Traveling salesman on N node complete graph(

αα+1

)ρ−1/αα 21/α

where C∗α is explicitly computed in Table 1.1. It is interesting that the asymp-

totic bounds exhibit the same scaling behavior (although with different con-stants) under the assumption of independence and that the limiting constantsfor the minimum spanning tree and the traveling salesman problems are thesame.

1.3. Structure of the paper. In section 2, we provide a general duality-basedapproach with which to compute the tight upper bound on the expected optimalobjective value of a 0-1 maximization problem under marginal moment information.Unfortunately, this formulation has exponentially many constraints in general. Insection 3, we obtain our first set of results based on the reformulation in [21] for theproblem. We introduce a semidefinite programming approach for solving a class ofsuch problems. In section 4, we provide numerical comparisons with other methodsfor a particular application in project management. In section 5, we develop thebound under complete and identical marginal distributions and provide new asymp-totic bounds for classical combinatorial problems. In section 6, we summarize ourconclusions.

2. General approach. Given limited marginal information for each objectivecoefficient ci, we are interested in computing the tight upper bound on the expectedoptimal objective value of a 0-1 maximization problem. Given the feasible set ofmarginal distributions in (1.5), we can rewrite problem (1.6) as

Z∗max = supθ Eθ[Z

∗max(c)]

s.t. Eθ[fik(ci)] = mik, k = 1, . . . , ki, i = 1, . . . , n,

Eθ[IΩ] = 1,

(2.1)

where Ω denotes Ω1 × · · · × Ωn. We construct the dual of this formulation by in-troducing a dual variable yik for each moment equality constraint and y00 for theprobability mass constraint. Since the primal problem is bounded, the dual problemis formulated as

Z∗D = min

(y00 +

n∑i=1

ki∑k=1

yikmik

)

s.t. y00 +

n∑i=1

ki∑k=1

yikfik(ci) ≥ Z∗max(c) ∀c ∈ Ω.

(2.2)

PROBABILISTIC COMBINATORIAL OPTIMIZATION 189

Theorem 2.1 (see Isii [14]). (a) The primal and dual formulations are relatedby weak duality as Z∗

max ≤ Z∗D.

(b) Furthermore, if the moments lie interior to the set of feasible moment vectorsfor arbitrary multivariate distributions, then strong duality holds, i.e., Z∗

max = Z∗D.

The strong duality condition in Theorem 2.1(b) is a Slater-type regularity con-dition that requires the moment vector to lie in the interior of the moment space.Under marginal moment specification, the multivariate moment space is defined asthe product of univariate moment spaces. Checking for the interior point conditionhence reduces to checking for the interior point condition for the univariate momentspaces. Such univariate conditions are easier to verify (as positive definiteness con-ditions on the moments matrix [15]). From this point on, we assume that this regu-larity condition is satisfied and hence the optimum objective value of problem (2.2),Z∗D = Z∗

max.

We now express the right-hand side of the constraint in formulation (2.2) explic-itly. Since the objective function Z∗

max(c) is a piecewise linear convex function, wecan rewrite the dual formulation as

Z∗max = min

(y00 +

n∑i=1

ki∑k=1

yikmik

)

s.t. y00 +

n∑i=1

ki∑k=1

yikfik(ci) −∑i∈Bp

ci ≥ 0 ∀c ∈ Ω, p = 1, . . . , P.

(2.3)

Each constraint in formulation (2.3) is equivalent to the nonnegativity of a multivari-ate function over a subset of n. For the simplest case with Ω defined by a discreteset of values, problems (2.1) and (2.3) reduce to the standard linear programmingprimal and dual problems. In general, there is an exponential number of such con-straints due to the exponential number of feasible solutions to the nominal problem(1.1). For the simplest case with fik(ci) = cki and Ω = n, each constraint reduces tothe nonnegativity of a multivariate polynomial over n. A simple sufficient conditionto ensure this is a sum of squares decomposition for the polynomial. This conditioncan be expressed by a positive semidefiniteness constraint on matrices of increasingsize in the dual variables (cf. [15] and [25]). As the order of the relaxation increases,the size of these matrices increases drastically. Moreover, tractable necessary con-ditions to ensure the nonnegativity of a multivariate polynomial are in general notknown. These reasons seem to suggest that formulation (2.3) is difficult to solve evenfor simple fik(·), regardless of the difficulty of the nominal problem.

3. Convex reformulation. In this section, we provide a reformulation for com-puting Z∗

max that uses the marginal information structure of the objective vector basedon a formulation originally proposed in [21] for a specific combinatorial problem undercomplete marginal distributions. Specifically, we apply the approach to general 0-1optimization problems.

3.1. Complete marginal distribution information. We first focus on thecase with completely known marginal distributions Fi for each objective coefficient,and we outline the approach of Meilijson and Nadas [21]. We denote by x+ =max(0, x). For each feasible solution to problem (1.1) indexed by p and for an arbi-

190 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

trary vector d ∈ n, we have∑i∈Bp

ci =∑i∈Bp

di +∑i∈Bp

(ci − di),

≤ maxp=1,...,P

∑i∈Bp

di +

n∑i=1

[ci − di]+,

= Z∗max(d) +

n∑i=1

[ci − di]+.

Since the right-hand side of the inequality is independent of any particular feasiblesolution of problem (1.1), we have

Z∗max(c) ≤ Z∗

max(d) +

n∑i=1

[ci − di]+.

Taking expectations with respect to Fi and the minimum over d ∈ n we obtain

Eθ[Z∗max(c)] ≤ min

d∈n

(Z∗

max(d) +

n∑i=1

EFi[ci − di]

+

)∀θ ∈ Θ(F1, . . . , Fn).

Furthermore, a joint probability distribution is constructed in [21] such that the upperbound with given marginal distributions Fi is tight. This brings them to their centralresult that solving a convex minimization problem in n variables yields the tight upperbound:

supθ∈Θ(F1,...,Fn)

Eθ[Z∗max(c)] = min

d∈n

(Z∗

max(d) +

n∑i=1

EFi[ci − di]

+

).(3.1)

3.2. Partial marginal distribution information. In this section, we considerthe case where the marginal distribution function Fi is not assumed to be known ex-actly but to lie in the set Fi. Optimizing over this set of feasible marginal distributions,and using (3.1), the tight upper bound Z∗

max is obtained by solving

Z∗max = sup

Fi∈Fi,i=1,...,nmin

d∈n

(Z∗

max(d) +

n∑i=1

EFi [ci − di]+

).

By interchanging the order of sup and min, we obtain an upper bound on Z∗max:

Z∗max ≤ min

d∈n

(Z∗

max(d) +

n∑i=1

supFi∈Fi

EFi [ci − di]+

).(3.2)

Klein Haneveld [13] showed that if the optimal marginal distribution F ∗i ∈ Fi for the

inner subproblem is the same for all di, that is,

supFi∈Fi

EFi [ci − di]+ = EF∗

i[ci − di]

+ ∀di ∈ , i = 1, . . . , n,(3.3)

then bound (3.2) is tight. We next give some examples of marginal moment informa-tion under which condition (3.3) is satisfied.

PROBABILISTIC COMBINATORIAL OPTIMIZATION 191

(a) Given bounded range Ωi = [ci, ci], since the function [ci − di]+ is a nonde-

creasing function in ci, the optimal marginal distribution F ∗i is

PF∗i(ci = ci) = 1.

Solving formulation (3.2) yields the tight upper bound as

Z∗max = Z∗

max(c).(3.4)

(b) Given bounded range Ωi = [ci, ci] and first moment µi for each coefficient ci,the optimal marginal distribution F ∗

i is known to be [20]

PF∗i(ci = ci) =

ci − µi

ci − ci, PF∗

i(ci = ci) =

µi − cici − ci

.

Since this distribution is independent of di, solving formulation (3.2) yields the tightbound Z∗

max. The decision vector d can be restricted to lie in the range Ω = [c, c]without affecting the optimal value. Hence the tight upper bound is the solution tothe linear program:

Z∗max = min

d∈[c,c]

(Z∗

max(d) +

n∑i=1

(µi − cici − ci

)(ci − di)

).(3.5)

(c) Given semi-infinite range Ωi = [ci,∞) and first moment µi, Birge and Mad-dox [5] extended the previous result to obtain

Z∗max = Z∗

max(c) +

n∑i=1

(µi − ci) .(3.6)

However, with additional second moment information, the optimal two-atom dis-tribution F ∗

i is dependent on the variable di [5]. Hence formulation (3.2) was notknown to be tight for higher order moment information. However, we next show thatit is in fact tight for any prescribed set of marginal moment information.

3.3. The main results. Let Z∗1 denote the optimum objective value of the

right-hand side of formulation (3.2). With prescribed marginal information in (1.5),this bound is computed as

Z∗1 = min

(Z∗

max(d) +

n∑i=1

supFiEFi [ci − di]

+

)

s.t. EFi [fik(ci)] = mik, k = 1, . . . , ki, i = 1, . . . , n,

EFi [IΩi ] = 1, i = 1, . . . , n.

(3.7)

Since no cross moment information is known, the ith inner subproblem is written as

supFiEFi

[ci − di]+

s.t. EFi [fik(ci)] = mik, k = 1, . . . , ki,

EFi[IΩi

] = 1.

To formulate the dual of the ith subproblem, we introduce variables yi0 for the prob-ability mass constraint and yik for the moment equality constraints. The dual of the

192 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

univariate subproblem is then written as

min

(yi0 +

ki∑k=1

yikmik

)

s.t. yi0 +

ki∑k=1

yikfik(ci) ≥ ci − di ∀ci ∈ Ωi,

yi0 +

ki∑k=1

yikfik(ci) ≥ 0 ∀ci ∈ Ωi.

Under the strong duality assumption, we substitute the dual of each subproblem intoformulation (3.7) to obtain the equivalent formulation:

Z∗1 = min

(Z∗

max(d) +

n∑i=1

yi0 +

n∑i=1

ki∑k=1

yikmik

)

s.t. pi1(ci) := yi0 + di +

ki∑k=1

yikfik(ci) − ci ≥ 0 ∀ci ∈ Ωi, i = 1, . . . , n,

pi2(ci) := yi0 +

ki∑k=1

yikfik(ci) ≥ 0 ∀ci ∈ Ωi, i = 1, . . . , n.

(3.8)

We denote by pi1(ci) and pi2(ci) the two univariate functions of ci that are nonnegativeover Ωi. There is a total of 2n such functions in formulation (3.8).

We next prove the equivalence of formulation (3.8) with the original dual (2.3),rewritten here for clarity:

Z∗max = min

(y00 +

n∑i=1

ki∑k=1

yikmik

)

s.t. y00 +

n∑i=1

ki∑k=1

yikfik(ci) −∑i∈Bp

ci ≥ 0 ∀c ∈ Ω, p = 1, . . . , P.

(3.9)

Theorem 3.1. The upper bound on the expected optimal objective value of the0-1 maximization problem Z∗

1 obtained by solving formulation (3.8) for a prescribedset of marginal moment information is tight, i.e., Z∗

1 = Z∗max.

Proof. Clearly, formulation (3.8) provides an upper bound on the solution ob-tained by solving formulation (3.9). Hence Z∗

max ≤ Z∗1 .

To show the reverse inequality, consider an optimal solution to formulation (3.9)represented by variables y∗00, y

∗ik. We now generate a feasible solution to formulation

(3.8) in the following manner. We set yik = y∗ik for all k and i with k, i ≥ 1. Havingfixed the variables yik, we solve problem (3.8) to optimality for the remaining variables.Let y∗i0 and d∗i be the corresponding optimal values for the remaining variables.

We first prove a minimality property of the nonnegative univariate functions informulation (3.8) that will be used later in the proof. We show that we can find anoptimal solution to this problem such that the value y∗i0 is minimal for pi2(ci) to benonnegative over Ωi. To see why, suppose that there is an ε > 0 such that we decreasey∗i0 by ε and pi2(ci) remains nonnegative over Ωi. Then we can increase d∗i by ε suchthat pi1(ci) remains unchanged. Since the objective function is an increasing function

PROBABILISTIC COMBINATORIAL OPTIMIZATION 193

in yi0, the modification in yi0 decreases the objective function by ε. Now, sinceZ∗

max(d) = maxp(∑

i∈Bpdi), the modification in di changes the objective function by

at most ε. Hence the above modification will not increase the objective function.Let y∗i0 be chosen such that it is the minimal value for pi2(ci) to be nonnegative

over the specified support. Similarly, having fixed y∗i0, we can decrease d∗i as longas pi1(ci) is nonnegative over Ωi since the objective is an increasing function in di.Hence we can restrict our attention to optimal values of y∗i0 and d∗i that are minimalfor the nonnegative functions pi1(ci) and pi2(ci) over Ωi.

In general, given n such univariate functions pi(ci) =∑ki

k=1 aikfik(ci) + ai0 withai0 at the minimal value for pi(ci) to be nonnegative over Ωi, the minimal value of

a00 for the multivariate function p(c) =∑n

i=1

∑ki

k=1 aikfik(ci)+a00 to be nonnegativeover Ω is

∑ni=1 ai0. To see this, we start with

pi(ci) =

ki∑k=1

aikfik(ci) + ai0 ≥ 0 ∀ci ∈ Ωi, i = 1, . . . , n,

and add the n functions to obtain

n∑i=1

ki∑k=1

aikfik(ci) +

n∑i=1

ai0 ≥ 0 ∀c ∈ Ω.

From the minimality of ai0 for the univariate function pi(ci), we know that thereexists an ci ∈ Ωi such that

pi(ci) =

ki∑k=1

aikfik(ci) + ai0 = 0, i = 1, . . . , n.

Adding these equalities, we obtain a vector c = (c1, . . . , cn) that lies in Ω, whichsatisfies

n∑i=1

ki∑k=1

aikfik(ci) +

n∑i=1

ai0 = 0.

Thus, the minimal value for a00 such that the function p(c) is nonnegative over Ω isa00 =

∑ni=1 ai0.

Now consider any feasible solution to problem (1.1) indexed by p. Then, fromformulation (3.8) we obtain

∀ i such that i ∈ Bp, y∗i0 + d∗i +

ki∑k=1

y∗ikfik(ci) − ci ≥ 0 ∀ci ∈ Ωi,

∀ i such that i /∈ Bp, y∗i0 +

ki∑k=1

y∗ikfik(ci) ≥ 0 ∀ci ∈ Ωi.

Summing up these n constraints, we obtain

n∑i=1

y∗i0 +∑i∈Bp

d∗i +

n∑i=1

ki∑k=1

y∗ikfik(ci) −∑i∈Bp

ci ≥ 0 ∀c ∈ Ω, p = 1, . . . , P.

194 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

From the minimality of the univariate functions, we know that∑n

i=1 y∗i0 +

∑i∈Bp

d∗iis the minimal value for this function to be nonnegative over Ω. Comparing this withthe multivariate function in formulation (3.9), we obtain

y∗00 ≥n∑

i=1

y∗i0 +∑i∈Bp

d∗i , p = 1, . . . , P,

which reduces to

y∗00 ≥n∑

i=1

y∗i0 + Z∗max(d

∗).(3.10)

From formulation (3.8) we obtain

Z∗1 ≤ Z∗

max(d∗) +

n∑i=1

y∗i0 +

n∑i=1

ki∑k=1

y∗ikmik

≤ y∗00 +

n∑i=1

ki∑k=1

y∗ikmik (from (3.10)),

= Z∗max.

By setting c := −c and d := −d, an equivalent result is obtained for Z∗min.

Corollary 3.2. The tight lower bound on the expected optimal objective valueof a 0-1 minimization problem for a prescribed set of marginal moment informationis obtained by solving

Z∗min = max

d∈n

(Z∗

min(d) +

n∑i=1

infFi∈Fi

EFi [min(0, ci − di)]

).(3.11)

3.4. Solution techniques. Since Theorem 3.1 implies that Z∗max = Z∗

1 , wefocus on solving formulation (3.8). We show that for a fairly general class of piecewisepolynomial functions fik, this formulation can be solved as a semidefinite program.Some examples of such functions are (ci−a)+ or I(−∞,a]. The power moment functions

fik(ci) = cki are included in this class.For each coefficient ci, the constraints in formulation (3.8) imply the nonnegativity

of two univariate functions, pi1(ci) and pi2(ci) over Ωi. For piecewise polynomialfunctions fik, we can decompose the support set Ωi to intervals Ωij , j = 1, . . . , li,with ∪jΩij = Ωi such that pi1(ci) and pi2(ci) are polynomials over each of theseintervals. We can now rewrite formulation (3.8) as

Z∗max = min

(Z∗

max(d) +

n∑i=1

yi0 +

n∑i=1

ki∑k=1

yikmik

)

s.t. pi1(ci) ≥ 0 ∀ci ∈ Ωij , j = 1, . . . , li, i = 1, . . . , n,

pi2(ci) ≥ 0 ∀ci ∈ Ωij , j = 1, . . . , li, i = 1, . . . , n,

(3.12)

where the constraints now correspond to the nonnegativity of univariate polynomialsover Ωij . If Ωij is a finite set of atoms, this reduces to linear constraints.

The key observation is that although it is difficult to express exactly the nonnega-tivity of a multivariate polynomial, we can use positive semidefinite constraints to ex-press the nonnegativity of a univariate polynomial over an interval. This simplification

PROBABILISTIC COMBINATORIAL OPTIMIZATION 195

arises due to the equivalence of the sum of squares representation and nonnegativityof a polynomial in the univariate case; see [23] and [4]. We focus on three specificintervals, namely, the entire real line Ω = (−∞,∞), the positive ray Ω = [0,∞), andthe segment Ω = [0, 1]. The semidefinite representation for all other intervals of can be obtained from simple affine transformations of these three cases.

Proposition 3.3 (see Nesterov [23] and Bertsimas and Popescu [4]). (a) A uni-

variate polynomial f(c) =∑2k

r=0 arcr is nonnegative over the interval Ω = (−∞,∞)

if and only if there exists a positive semidefinite matrix Y = [Yij ]i,j=0,...,k such that

ar =∑

i,j: i+j=r

Yij , r = 0, . . . , 2k,

Y 0.

(b) A univariate polynomial f(c) =∑k

r=0 arcr is nonnegative over the interval

Ω = [0,∞) if and only if there exists a positive semidefinite matrix Y = [Yij ]i,j=0,...,k

such that

0 =∑

i,j: i+j=2r−1

Yij , r = 1, . . . , k,

ar =∑

i,j: i+j=2r

Yij , r = 0, . . . , k,

Y 0.

(c) A univariate polynomial f(c) =∑k

r=0 arcr is nonnegative over the interval

Ω = [0, 1] if and only if there exists a positive semidefinite matrix Y = [Yij ]i,j=0,...,k

such that

0 =∑

i,j: i+j=2r−1

Yij , r = 1, . . . , k,

r∑l=0

(k − l

r − l

)al =

∑i,j: i+j=2r

Yij , r = 0, . . . , k,

Y 0.

Clearly, from Proposition 3.3, we can replace the condition that a univariatepolynomial is nonnegative over an interval of by a requirement on a matrix to bepositive semidefinite and a set of linear equalities that must be satisfied. The specificform depends on the nature of the interval Ω. For example, from Proposition 3.3(a),for a univariate polynomial to be nonnegative over , the degree of the polynomialmust be even.

From Proposition 3.3, we can rewrite formulation (3.12) as

Z∗max = min

(Z∗

max(d) +

n∑i=1

yi0 +

n∑i=1

ki∑k=1

yikmik

)

s.t. Wij(yi0, . . . , yiki ,−1, di,Ωij) 0, j = 1, . . . , li, i = 1, . . . , n,

Xij(yi0, . . . , yiki ,Ωij) 0, j = 1, . . . , li, i = 1, . . . , n,

(3.13)

where Wij and Xij denote the semidefinite constraint matrix Y obtained from Propo-sition 3.3. The entries of the semidefinite matrix depends on the nature of the intervalΩij and the coefficients of the polynomial.

196 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

Theorem 3.4. Given marginal distribution information as moments of piecewisepolynomial functions, the tight upper (lower) bound on the expected optimal objectivevalue of a 0-1 maximization (minimization) problem can be computed in polynomialtime if the nominal problem can be solved in polynomial time.

Proof. Formulation (3.13) can be reformulated as

Z∗max = min

(t +

n∑i=1

yi0 +

n∑i=1

ki∑k=1

yikmik

)

s.t. Wij(yi0, . . . , yiki ,−1, di,Ωij) 0, j = 1, . . . , li, i = 1, . . . , n,

Xij(yi0, . . . , yiki,Ωij) 0, j = 1, . . . , li, i = 1, . . . , n,

t ≥ Z∗max(d),

(3.14)

where t is an additional variable. We have a polynomial number of polynomial-size semidefinite matrices in this formulation. One approach to solve this prob-lem is to disaggregate the term Z∗

max(d) in terms of its feasible solutions. How-ever, the number of linear constraints in such a formulation would be exponen-tially large. This implies that solving formulation (3.14) is still difficult in general.For the class of polynomially solvable 0-1 maximization problems, we can, however,solve this efficiently by considering the separation version of the problem. Since thesemidefinite constraints are polynomial sized, this essentially reduces to the sepa-ration problem: Given t and d, verify if t ≥ Z∗

max(d), and if not find a violatedinequality.

To solve the separation problem, we solve the nominal discrete optimization prob-lem with objective vector d. If this can be solved in polynomial time, then we can findthe optimal decision vector indexed by p∗ ∈ 1, . . . , P and Z∗

max(d) in polynomialtime. Then we simply test if t ≥ Z∗

max(d), and if not an index p∗ is found such that∑i∈Bp∗

di > t. The desired result follows from the equivalence of separation and

optimization [11].

Remarks. (a) Theorem 3.4 implies that the tight lower bounds for the expectedoptimal objective values of combinatorial problems such as the minimum spanningtree, assignment, matching, and shortest path problems are computable in polynomialtime under the marginal moment model.

(b) The previously known results (3.4), (3.5), and (3.6) can be viewed as specialcases of our Theorems 3.1 and 3.4.

(c) The results hold even if we only know upper bounds on moments on E[fik(ci)].The only difference is that the corresponding dual variables are nonnegative.

(d) The variables di in formulation (3.14) can be restricted to lie in [ci, ci], whereci and ci define lower and upper bounds on the range of Ωi.

The solution method described thus far is a general cutting plane algorithm thatsolves the nominal problem (1.1) at each step in conjunction with semidefinite con-straints. We now provide a compact semidefinite reformulation for a class of 0-1optimization problems that may be useful computationally.

Theorem 3.5. For 0-1 optimization problems with compact linear programmingformulations, the tight upper (lower) bound on the expected optimal objective value ofa 0-1 maximization (minimization) problem under the marginal moment model canbe computed as a compact semidefinite program.

Proof. Any 0-1 optimization problem can be reformulated as a linear programming

PROBABILISTIC COMBINATORIAL OPTIMIZATION 197

problem over the convex hull of its feasible region:

Z∗max(d) = max d′x

s.t. Ax ≤ b,(3.15)

where (A, b) provides the convex hull representation. Using linear programming du-ality we have

Z∗max(d) = min p′b

s.t. p′A = d′,

p ≥ 0.

(3.16)

Substituting this dual into formulation (3.13), we obtain

Z∗max = min

(p′b +

n∑i=1

yi0 +

n∑i=1

ki∑k=1

yikmik

)

s.t. Wij(yi0, . . . , yiki,−1, di,Ωij) 0, j = 1, . . . , li, i = 1, . . . , n,

Xij(yi0, . . . , yiki,Ωij) 0, j = 1, . . . , li, i = 1, . . . , n,

p′A = d′,

p ≥ 0.

(3.17)

The convex hull (A, b) representation could be exponentially large or not knownexplicitly. However, for a class of problems this linear representation is compact.For such problems, formulation (3.17) is a compact semidefinite program that can besolved efficiently in polynomial time.

Examples of 0-1 optimization problems with compact linear programming repre-sentation include the assignment and network flow problems such as the shortest pathproblem. The longest path problem on acyclic graph is also an example of 0-1 opti-mization problem with compact LP formulation, although the longest path problemis NP-hard in general.

4. Application in project management. In this section, we provide an ap-plication of the techniques developed to the problem of computing the longest path ina directed acyclic graph. Such problems arise in project management and scheduling.

4.1. Description of the problem. A project is specified as a set of activitiesthat has to be completed given certain precedence relationships. We represent aproject in an activity-on-arc framework with an acyclic directed graph G(V ∪s, t, E),where |E| = n arcs. An arc in this graph represents an activity, and a node representsthe completion of all activities leading to this node. Node s represents the start ofthe project while node t represents the completion of the project. The precedenceconstraints in the graph imply that if there exist arcs (i, j) and (j, k) in the graph,then job (i, j) must be performed before job (j, k). Such a graph constructed isnecessarily acyclic, otherwise it would lead to an inconsistent ordering of jobs. Welet the nonnegative random arc lengths cij denote the time required to complete eachindividual activity. Given this setup, the parameter of importance is the time requiredto complete the project measured by the longest path from the start node s to theend node t in the graph. The longest path in the graph Z∗

max(c) is computed by

198 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

solving

Z∗max(c) = max

∑(i,j)∈E

cijxij

s.t.∑

j:(i,j)∈E

xij −∑

j:(j,i)∈E

xji =

⎧⎨⎩

1 if i = s,−1 if i = t,0 if i ∈ V ,

xij ∈ 0, 1 ∀(i, j) ∈ E.

(4.1)

In fact, we will focus on a more general measure of project performance, called projecttardiness. Assume that we are specified a due date T for the project. Project tar-diness is defined as a linear cost that is incurred only if the project completion timeZ∗

max(c) exceeds the due date. If the project is completed before the deadline, thenno cost is incurred. Mathematically, it is expressed as a piecewise convex function ofproject completion time:

Project tardiness G(T) := [Z∗max(c) − T ]+.

It is clear that for T = 0, G(0) reduces to Z∗max(c).

Computing the longest path in a general graph is NP-hard, while it is polyno-mially time solvable for acyclic graphs [16]. However, if the arc lengths are ran-dom variables that are distributed independently and restricted to two possible val-ues, Hagstrom [12] has shown that the computation of the expected longest pathis #P-hard. Furthermore, the expected longest path cannot be computed in timepolynomial in the number of values the completion time takes unless P = NP . Thissuggests that in general it is difficult to compute the exact expected project tardinessunder uncertainty in activity duration, hence motivating the interest in bounds andapproximations.

Robillard and Trahan [26] studied the expected completion time E[Z∗max(c)] under

assumptions of complete knowledge of distribution of activity durations and indepen-dence. Assuming independence, but only limited moment information, Devroye [8]computed upper bounds on the first moment and second moments of the comple-tion time. However, Levy and Wiest [18] argued that activity durations are oftendependent due to resource limitations. It is, however, not practical to estimate thiscomplete multivariate joint distribution. Hence there has been effort focused on esti-mating the expected case tardiness under knowledge of limited moment informationof the individual activities.

One approach to this problem deals with approximating the expected projecttardiness. Under specific assumptions on the moments, the central limit theorem hasbeen used to approximate the distribution of the project completion time; see [2], [27],and [19]. Under this method, the completion times obtained from different paths inthe network are assumed to obey a multivariate normal distribution. Then, evaluatingthe expected project tardiness involves calculating the maximum of correlated normaldistributions, a generally nontrivial calculation. Furthermore, if the various activitiesare correlated, one needs to impose restrictions on the distribution of arc lengths forthe central limit theorem to apply. We do not make any such assumptions in thispaper. Finally, even if the central limit theorem can be applied, it will not be a goodapproximation for smaller networks.

An alternative approach to the problem focuses on computing bounds on theexpected project tardiness. A simple lower bound can be computed by applying

PROBABILISTIC COMBINATORIAL OPTIMIZATION 199

1 10

9

8

7

6

2

5

4

3

Fig. 4.1. Project network in Example 1.

Jensen’s inequality to the convex tardiness function. Computing tight upper boundsis more challenging and more important as it provides an estimate of the worst caseperformance. Klein Haneveld [13] provided a tight upper bound on E[G(T )] whenthe first moments of activity durations are given. With additional second momentinformation, Birge and Maddox [5] used the formulation in [21] to compute the upperbound on expected tardiness. However, as the computational results indicate, anapproximation of the original objective function is used to compute this upper bound.The lack of precision from using such a linearization approach for solving a nonlinearproblem demonstrates an additional advantage of using the semidefinite optimizationapproach. It should be noted that while formulation (3.13) can be directly used tocompute the worst case expected project completion time, it could be easily extendedto the tardiness objective. Specifically, we need to replace the term Z∗

max(d) in theobjective function by [Z∗

max(d)−T ]+ and solve the corresponding semidefinite programto compute the worst case expected tardiness denoted as G∗(T ):

G∗(T ) = min

([Z∗

max(d) − T ]+ +

n∑i=1

yi0 +

n∑i=1

ki∑k=1

yikmik

)

s.t. Wij(yi0, . . . , yiki,−1, di,Ωij) 0, j = 1, . . . , li, i = 1, . . . , n,

Xij(yi0, . . . , yiki,Ωij) 0, j = 1, . . . , li, i = 1, . . . , n.

(4.2)

Since Z∗max(d) can be computed in polynomial time, formulation (4.2) is solvable in

polynomial time.

4.2. Computational examples. We consider two sample projects taken from[5]. The specified data are as follows:

(a) The minimum time required to complete each activity is known. We assumethat the maximum time required to complete an activity is infinity.

(b) For each activity, the first and second moments of the time to complete itare known. In addition for the first project, third moment information isassumed to be known. We add in this information to check the value of theextra information on the upper bound.

Under this marginal moment information, we solve formulation (4.2) with SeDuMiversion 1.03 developed by Sturm [29]. An interior point method is used by the softwareto solve the semidefinite program. The computations were conducted on a Pentium II(550 MHz) Windows 2000 platform and the total computation time was less than aminute.

Example 1. The first project depicted in Figure 4.1 consists of 10 activities thatare distributed over 3 paths. The marginal distribution information for the activities

200 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

Table 4.1

Activity duration data for Example 1.

Arc Range First mom. Second mom. Third mom.1 [1,∞) 1.0 1.0 1.02 [4,∞) 3.0 9.333 30.03 [3,∞) 2.0 4.333 10.04 [3,∞) 2.5 6.333 16.255 [7,∞) 5.0 26.333 146.06 [5,∞) 4.0 16.333 68.07 [5,∞) 3.0 10.333 40.08 [5,∞) 4.5 20.333 92.259 [2,∞) 1.5 2.333 3.7510 [6,∞) 5.0 25.333 130.0

Table 4.2

Upper bounds on expected tardiness for Example 1.

Information Bound/ T 0 15 18.33 21.67 25Range, first, second mom. G∗(T ) reported in [5] 20.35 5.35 2.98 1.27 0.73Range, first, second mom. G∗(T ) from eq. (4.2) 20.29 5.29 2.66 0.94 0.47Range, first, second, third mom. G∗(T ) from eq. (4.2) 20.10 5.10 2.53 0.75 0.18

27 29

28

22

20

23

21

14

13

18

2

3

1

4

11

9

6

5

19

8 15 16 17

25

10

12

26

24

7

Fig. 4.2. Project network in Example 2.

is provided in Table 4.1. We consider five different deadline times T . First, wecompute the worst case expected tardiness with range, first, and second momentinformation and compare it with the results reported in [5]. The results are providedin Table 4.2. In light of Theorem 3.1, it is clear that in [5] a heuristic method is used tocalculate the bound. We next calculate the tight upper bound with additional thirdmoment information. It is noted from Table 4.2 that the bounds are considerablytighter, especially for larger deadline dates T , indicating the value that additionalinformation has on these estimates. It is important to note that our semidefiniteprogramming formulation provides an easy way to incorporate third and higher ordermoment information, whereas previous techniques could not handle this.

Example 2. The second project is a larger one with 29 activities that are dis-tributed over 14 paths. The activity-on-arc graph representation of the project isprovided in Figure 4.2 and the data are provided in Table 4.3. The deadline time Tis varied from 45.10 to 85 approximately in steps of 10, and the worst case expected

PROBABILISTIC COMBINATORIAL OPTIMIZATION 201

Table 4.3

Activity duration data for Example 2.

Arc Range First mom. Second mom. Arc Range First mom. Second mom.1 [6,∞) 8.0 66.778 16 [3,∞) 4.0 16.1112 [14,∞) 17.0 298.0 17 [8,∞) 10.0 100.4443 [14,∞) 17.0 298.0 18 [1,∞) 2.0 4.1114 [2,∞) 4.0 17.0 19 [6,∞) 8.0 65.05 [1,∞) 2.0 4.444 20 [1,∞) 2.0 4.1116 [1,∞) 2.0 4.111 21 [6,∞) 8.0 65.07 [0.4,∞) 0.6 0.370 22 [1,∞) 2.0 4.1118 [0.4,∞) 0.6 0.370 23 [6,∞) 8.0 66.7789 [2,∞) 3.0 9.250 24 [1,∞) 2.0 4.11110 [2,∞) 3.0 9.250 25 [4,∞) 6.0 36.44411 [3,∞) 4.0 16.111 26 [1,∞) 2.0 4.11112 [3,∞) 4.0 16.111 27 [0.1,∞) 0.5 0.25713 [3,∞) 4.0 16.111 28 [2,∞) 4.0 16.44414 [3,∞) 4.0 16.111 29 [1,∞) 2.0 4.11115 [2,∞) 3.0 9.250

Table 4.4

Upper bounds on expected tardiness for Example 2.

Information Bound/ T 45.10 55 65 75 85Range, first, second mom. G∗(T ) reported in [5] 6.77 2.51 1.72 1.04 0.73Range, first, second mom. G∗(T ) from eq. (4.2) 6.52 2.38 1.38 0.97 0.73

project tardiness is computed for these times. The computed values are provided inTable 4.4 along with the values reported in [5]. Clearly these results indicate that thesemidefinite programming method provides a general tractable approach for comput-ing the worst case expected tardiness under moment information.

5. Asymptotic performance. In this section, we study the properties of thebounds asymptotically as the size of the problem increases to infinity. We make thefollowing assumptions:

(A) Each objective coefficient is a random variable with a common prespecifieddistribution function F . For simplicity, we assume that F is a continuousdistribution. No assumption of independence is made.

The complete knowledge of F is an important one in this setting to obtain in-teresting scaling results. Under this setting, the tight upper bound on the expectedoptimal value of a 0-1 maximization problem is computed as

Z∗max = supθ Eθ[Z

∗max(c)]

s.t. θ ∈ Θ(F, . . . , F ).(5.1)

We restrict ourselves to a fairly general class of a 0-1 optimization problems thatsatisfy the following two properties:

(B) All feasible solutions to problem (1.1) indexed by p ∈ 1, . . . , P have thesame cardinality. If we denote this common cardinality as K1, we have

K1 = |i : i ∈ Bp| ∀p = 1, . . . , P,

where |S| denotes the cardinality of set S.(C) Every element of the ground set X is contained in the same number of feasible

solutions. If we denote this common value as K2, we have

K2 = |p : i ∈ Bp| ∀i = 1, . . . , n.

202 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

Table 5.1

Parameters for three classical combinatorial optimization problems.

0-1 Optimization problem n K1 P K2

N ×N Linear assignment N2 N N ! (N − 1)!

Spanning tree on N node complete graph N(N−1)2

N − 1 NN−2 2NN−3

Traveling salesman on N node complete graph N(N−1)2

N (N−1)!2

(N − 2)!

Clearly K1 ≤ n and K2 ≤ P . In conjunction, these assumptions imply thatK1 and K2 are related by

K1P = K2n.

Three classical combinatorial problems that satisfy properties (b) and (c) are asfollows:

(a) Linear assignment problem: given a set 1, 2, . . . , N and an N×N objectivecoefficient matrix [cij ], find a permutation φ of the set that optimizes the sum∑N

i=1 ciφ(i).(b) Spanning tree problem on a complete graph: given a complete undirected

graph G(V,E) with N vertices and an N ×N symmetric objective coefficientmatrix [cij ], find a spanning tree with the optimal sum of the edge coefficients.

(c) Symmetric traveling salesman problem on a complete graph: given a completeundirected graph G(V,E) with N vertices and an N×N symmetric objectivecoefficient matrix [cij ], find a cyclic permutation φ of the set of N vertices

that optimizes∑N

i=1 ciφ(i).The parameters for these three problems are summarized in Table 5.1.Remarks. (a) While the first two problems are solvable in polynomial time, the

traveling salesman problem is difficult to solve. All three problems, however, sharestructural properties (B) and (C). The asymptotic results in this section in fact donot depend on the complexity of the nominal optimization problem.

(b) An example of a problem that does not satisfy properties (B) and (C) is theSteiner tree problem.

Theorem 5.1. Under assumptions (A)–(C), the tight upper bound on the ex-pected optimal objective value of a 0-1 maximization problem is

Z∗max = K1EF

[c∣∣∣ c ≥ F−1

(1 − K1

n

)],(5.2)

where F−1 is the inverse distribution function for F .Proof. From the previous sections, we know that tight bound Z∗

max on the ex-pected value of a 0-1 maximization problem given marginal distribution F is obtainedby solving formulation (3.1). Problem (5.1) can then be solved as

Z∗max = min

d∈n

(Z∗

max(d) +

n∑i=1

EF [ci − di]+

).(5.3)

Formulation (5.3) is a convex minimization problem in n variables. The Karush–Kuhn–Tucker conditions provide the necessary and sufficient optimality conditionsfor this problem. These conditions are

(i) λp ≥ 0 for all p = 1, . . . , P ,

PROBABILISTIC COMBINATORIAL OPTIMIZATION 203

(ii)∑P

p=1 λp = 1,(iii) Z∗

max(d) ≥∑

i∈Bpdi and λp = 0 whenever Z∗

max(d) >∑

i∈Bpdi for all p =

1, . . . , P , and(iv) 1 −

∑p:i∈Bp

λp = F (di) for all i = 1, . . . , n.We now generate a set of primal di and dual variables λp that satisfy conditions(i)–(iv).

Set the dual multipliers to λp = 1/P, p = 1, . . . , P . Clearly this satisfies conditions(i) and (ii). From (C), condition (iv) then reduces to

1 − K2

P= F (di) ∀i = 1, . . . , n.

If F−1 denotes the inverse of the distribution function, then the primal variables areset to

di = F−1

(1 − K2

P

)∀i = 1, . . . , n

or equivalently

di = F−1

(1 − K1

n

)∀i = 1, . . . , n.

Assumption (B) implies that Z∗max(d) = K1d, satisfying condition (iii). Thus we have

the optimal primal and dual variables to formulation (5.3). Substituting the optimalvalue of di into the objective function yields

Z∗max = K1F

−1

(1 − K1

n

)+ nEF

[c− F−1

(1 − K1

n

)]+

,

= K1F−1

(1 − K1

n

)+ nEF

[c− F−1

(1 − K1

n

) ∣∣∣ c ≥ F−1

(1 − K1

n

)]K1

n,

which reduces to the required result.Corollary 5.2. Under assumptions (A)–(C), the tight lower bound on the

expected optimal objective value of a 0-1 minimization problem is

Z∗min = K1EF

[c∣∣∣ c ≤ F−1

(K1

n

)].(5.4)

5.1. Applications. We evaluate the explicit expected optimal objective valuefor a class of distributions next. Note that in the case K1/n ↓ 0 as n ↑ ∞, c ≥ 0,and from (5.4), the asymptotic performance of Z∗

min depends only on the distributionfunction of the cost coefficient F (c) as c ↓ 0. We thus focus on 0-1 minimizationproblems with the assumption that the distribution function of the cost coefficientsFα satisfies

Fα(c) = ραcα as c ↓ 0.(5.5)

The density function fα is respectively assumed to satisfy

fα(c) = ρααcα−1 as c ↓ 0.(5.6)

Distributions of this type have been extensively studied in the asymptotic analysisof combinatorial problems [1], [3]. Models with distributions that satisfy (5.5) includethe following:

204 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

(a) Uniformly distributed cost coefficients U [0, 1] with α = 1 and ρα = 1;(b) Euclidean model with Fα defined as the distribution of the distances between

independently and uniformly distributed points in dimension α; here ρα rep-resents the volume of a sphere of unit radius in dimension α; and

(c) Independent model with the cost coefficients independently distributed withdistribution F .

We generalize these models by allowing the cost coefficients to be dependent.Under this distributional assumption we obtain the following result.

Theorem 5.3. Under assumptions (A)–(C) with cost distribution satisfying(5.5), the tight lower bound on the expected optimal objective value of a 0-1 min-imization problem that satisfies K1 = Θ(

√n) scales asymptotically to a (positive)

constant Cα as

limn→∞

(Z∗

min

n(α−1)/2α

)= Cα,(5.7)

where the asymptotic constant Cα depends on the nominal problem.

Proof. For cost coefficients with distribution function in (5.5), we obtain

F−1α (c) =

(c

ρα

)1/α

as c ↓ 0.

The tight lower bound Z∗min from Corollary 5.2 is then computed as

Z∗min = K1EFα

[c∣∣∣ c ≤ (

K1

ραn

)1/α],

= K1

⎡⎣∫ ( K1

ραn )1/α

0

ρααcαdc

⎤⎦/⎡

⎣∫ ( K1ραn )

1/α

0

ρααcα−1dc

⎤⎦ as n ↑ ∞.

Note that with K1 = Θ(√n), the term (K1/ραn)1/α ↓ 0 as n ↑ ∞. Evaluating the

integrals explicitly, we obtain

Z∗min =

(α

α + 1

)K1

(K1

nρα

)1/α

as n ↑ ∞,(5.8)

which brings us to the scaling result.

Corollary 5.4. The tight lower bound on the expected optimal objective value ofthe linear assignment, spanning tree, and traveling salesman problem with the marginaldistribution satisfying (5.5) scales asymptotically to a (positive) constant C∗

α as

limN→∞

(Z∗

min

N (α−1)/α

)= C∗

α,(5.9)

where the asymptotic constant C∗α depends on the nominal problem.

Proof. Using (5.8) and N = Θ(√n) for the three combinatorial problems of

interest, we obtain the desired scaling result.

We evaluate this asymptotic behavior for three specific problems in more detailnext.

PROBABILISTIC COMBINATORIAL OPTIMIZATION 205

Table 5.2

Asymptotic scaling constant for the linear assignment problem.

Parameter α With independence: C∗α Without independence: C∗

α

2 0.7535 0.66673 0.8953 0.75004 0.9474 0.80005 0.9602 0.83336 0.9742 0.8571

Minimum cost linear assignment problem. (a) Uniform distribution. Therandom minimum cost linear assignment problem has been extremely well studiedunder the uniform distribution model on [0, 1]. Under the additional assumption ofindependence, Walkup [30] proved an upper bound of 3 on the expected objectivevalue. The first proven lower bound for this expected value was 1.368 [17]. Thesebounds have respectively been strengthened to 1.94 in [7] and 1.51 in [24]. Aldous [1]more recently proved a conjecture of Mezard and Parisi [22] that under this modelthe expected optimal objective value approaches π2/6 ≈ 1.645 as the size approaches∞. In fact, this limiting value holds for any nonnegative continuous distribution suchthat the density of independent costs is 1 at 0. All of the above results hold underthe strict assumption of independence.

Without this assumption, by setting α = 1, ρα = 1, K1 = N , and n = N2 in(5.8), we obtain the tight lower bound as

For the linear assignment problem, Z∗min = 0.5 ∀N.(5.10)

Thus, by dropping the assumption of independence, the lower bound for the linearassignment problem becomes C∗

1 = 0.5.

(b) Power density. The linear assignment problem has been studied under theassumption that the cost coefficients are distributed on [0, 1] with density αcα−1 forα > 1. Under the assumption of independence, Donath [9] showed that asymptoticallythe expected optimal objective value is bounded as

C∗α ≤ lim

N→∞

(E[Z∗

min(c)]

N (α−1)/α

)≤ C∗

α

(α− 1

α

)

for some constant C∗α. The values of this constant for small values of α are provided

by the same author.

By setting K1 = N and n = N2 in (5.8), we obtain

For the linear assignment problem,

(Z∗

min

N (α−1)/α

)=

(α

α + 1

)∀N.(5.11)

Clearly, the tight lower bound scales with respect to N (α−1)/α under our model, too.This is similar to the scaling behavior observed under the independent model. InTable 5.2, we compare the constant C∗

α reported from [9] with C∗α = α/(α + 1) for

five values of α. As should be expected, C∗α is smaller than C∗

α.

For these two particular distributions it can be verified that our results for thelinear assignment problem hold for all N and are not just an asymptotic result. Thisis in contrast to the independent distribution model.

206 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

Minimum spanning tree and traveling salesman problems. (a) Uniformdistribution. The random minimum spanning tree problem on a complete graph hasbeen studied under the uniform distribution model. Under the additional assumptionof independence, Frieze [10] showed that the expected length of the minimum spanningtree asymptotically converges to ζ(3) ≈ 1.202.

Without making the assumption of independence on arc length distribution, weobtain with K1 = N − 1 and n = N(N − 1)/2 from (5.8):

Z∗min = 1 − 1

Nas N ↑ ∞.

As the size of the problem N approaches infinity, the tight lower bound clearly con-verges to C∗

1 = 1:

For the spanning tree problem, limN→∞

Z∗min = 1.(5.12)

This value is surprisingly close to the expected optimal objective value under inde-pendence. For the traveling salesman problem, we similarly obtain

Z∗min = 1 +

1

1 −Nas N ↑ ∞

and the same asymptotic constant of 1,

For the traveling salesman problem, limN→∞

Z∗min = 1.(5.13)

(b) Euclidean and independent model. For the independent model with distribu-tion function satisfying (5.5), the expected optimal objective value of the spanningtree asymptotically scales as [3]

limN→∞

(E[Z∗

min(c)]

N (α−1)/α

)=

1

αρ1/αα

∞∑k=1

Γ(k + 1d − 1)

k!k1/d+1.

Using the same model but without assuming independence, we obtain the tight lowerbound for the spanning tree problem as

limN→∞

(Z∗

min

N (α−1)/α

)=

(α

α + 1

)ρ−1/αα 21/α.(5.14)

Under the Euclidean model we can set the value of ρα = πd/2/Γ(d/2 + 1). Thevalues of the limiting constants are provided in Table 5.3. Interestingly, the travelingsalesman problem exhibits the same asymptotic scaling constant but the rates ofconvergence are different.

5.2. Limited marginal information. The previous section showed that with-out the assumption of independence, the asymptotic performance for a general classof combinatorial optimization problems exhibits interesting behavior under the modelof completely known marginal distributions. We now show that under the marginalmoment model such problems in fact may exhibit a different kind of scaling behavior.

Consider probabilistic combinatorial optimization problems that satisfy assump-tions (B) and (C), and the following:

(A′) Each objective coefficient is a random variable with common first and secondmoments µ and µ2 + σ2.

PROBABILISTIC COMBINATORIAL OPTIMIZATION 207

Table 5.3

Asymptotic scaling constant for the spanning tree problem.

Parameter α With independence: C∗α Without independence: C∗

α

2 0.5684 0.53193 0.6094 0.58624 0.6558 0.63835 0.7010 0.68676 0.7440 0.7317

(D) The ground set can be partitioned into disjoint collection of feasible solutions,i.e., there is a set 0, . . . , P − 1 such that

X = ∪p=0,...,P−1Bp, Bi ∩Bj = ∅ ∀i = j, i, j = 0, . . . , P − 1.

Note that we change the indexing of the feasible solutions here slightly forease of developing the result.

Assumption (D) is valid for a wide variety of combinatorial optimization prob-lem. For instance, for the assignment problem, (D) follows from the well-known factthat the edges of a complete bipartite graph KN×N can be partitioned into a collec-tion of N disjoint perfect matching. For the spanning tree and traveling salesmanproblems, assumption (D) is obtained for an even and an odd number of vertices N ,respectively.

Under these assumptions, we show that the tight lower bound Z∗min = 0 for these

problems. To see this, we construct an instance of the problem, and a correspond-ing distribution satisfying all the conditions, such that the expected minimum costsolution is zero.

Let B0, B1, . . . , BP−1 be the feasible solutions stipulated by assumption (D).We define

m =

(µ2

µ2 + σ2

)P ,

Sk = ∪k+m−1i=k Bi, k = 0, . . . , P − 1,

where the addition is taken modulo P . Consider the cost function ck, with

cki =

⎧⎨⎩

(µ2 + σ2

µ

)if i ∈ Sk,

0 otherwise.

Note that for a particular objective function ck, since there is a feasible solutionBl, l /∈ Sk, with zero optimal objective value, ck(Bl) = 0. Construct a probabilitydistribution with c = ck with probability 1/P . For this distribution it can be verifiedthat the first two moments are

E[ci] =m

P

(µ2 + σ2

µ

)= µ, i = 1, . . . , n,

E[c2i ] =m

P

(µ2 + σ2

µ

)2

= µ2 + σ2, i = 1, . . . , n.

Furthermore, for each realization of c, the optimum solution value is 0, so we haveZ∗

min = 0.

208 D. BERTSIMAS, K. NATARAJAN, AND C.-P. TEO

6. Conclusions and extensions. In this paper, we addressed the problem ofcomputing tight bounds on the expected value of a 0-1 optimization problem un-der uncertainty in the objective coefficients. The feasible multivariate distributionswere characterized by known moment information on the marginal distributions ofthe objective coefficients. We showed that given moment information on piecewisepolynomial functions of the objective coefficients, the tight upper (lower) bound onthe expected optimal objective value of the 0-1 maximization (minimization) problemcan be computed in polynomial time if the deterministic problem is solvable in poly-nomial time. We provided an efficiently solvable semidefinite program to compute thistight bound. Under an extension of this model with completely known and identicalmarginal distributions we analyzed the asymptotic bounds for a class of combinatorialproblems.

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